Young Children
The Mishnah Berurah (561:17) brings from the Magen Avraham that a young child is exempt from performing keriah. The Steipler explains (Orchos Rabbeinu, Vol. 2, pg. 154) that the mitzvah of chinuch doesn’t apply by this mitzvah, and although we perform keriah for young children when a relative dies (Shulchan Aruch, Yoreh Deah 340:27) that is because of agmas nefesh (see Rashi, Moed Kotan 14b who explains, the onlookers will cry when they see this, and it is kavod for the person who died), however, this is not applicable when performing keriah by the Kosel HaMaravi.
As to why there is no chinuch by this mitzvah, it is said over from R’ Elyashiv that it is because we hope that by the time the child gets older, the Beis HaMikdosh will be rebuilt.
However, from the Netziv (Ha’amek Sheilah, Sheilta 35, s.k. 13) it seems that young children are obligated to perform keriah.
Women
The minhag of many women today is that they don’t perform keriah when seeing the Makom HaMikdosh (this is very difficult, as what is the source for this, and why should it be different to performing keriah when a relative passes away). However, the Steipler (Orchos Rabbeinu, Vol. 2, pg. 149) held that women are obligated just as much as men, and he would tell his wife to perform keriah. This was also the ruling of R’ Sroya Davlitsky (Zichron Betzalel pg. 124).
After a woman performs keriah, she should join the tear together with safety pins due to tznius (Menuchas Emes, Perek 12, he’orah 15). What’s even better is if she wears a separate garment just for the mitzvah of keriah, and then afterwards she removes it, and wears her regular untorn clothes.
A Son Who Is Still Being Supported by His Father
The Shu”t L’Horas Nossan (2:39) rules that a son who is still being supported by his father must perform keriah. Although the Shulchan Aruch (Hilchos Semachos, Yoreh Deah 340:34) rules that if one borrows a garment from his friend it is forbidden to tear it, even if one’s father died (and for this reason, soldiers that come to the Makom HaMikdosh with uniform don’t perform keriah, as the uniform doesn’t belong to them), and in our case even if the son tells his father that he is going to the Kosel HaMaravi, who says the father is mochel [forgiving] and lets him tear the clothes, nonetheless It is allowed as what a father gives his son is considered tzedokah and belongs to the son. Additionally, a father is happy when his son does a mitzvah. This is also the ruling of R’ Moshe Sternbuch (Shu”t Teshuvos V’Hanhagos, Vol. 5, end of 165) and R' Yisroel Yaakov Fisher (Halichos Even Yisroel, pg. 393).
Chosan and Kallah During the Week of Shevah Berachos
R’ Elyashiv zt”l ruled that a chosan and kallah during the week of shevah berachos are exempt from performing keriah, as it is considered a Yom Tov for them (cited in Menashim B’Ohel, pg. 154). See the Chochmas Adam (Klal 154, 4) who brings from the Maharil in Hilchos Aveilus, that a chosan doesn’t perform keriah during the week of shevah berachos (see later about Yom Tov and Chol HaMoed).
Oinen
An oinen who travels to bury his relative on Har HaZeisim, and on the way he sees the Makom HaMikdosh he is exempt from performing keriah. And after the burial he doesn’t need to perform keriah either, as he was already exempt at the beginning (Zichron Betzalel, pg. 124).
A Blind Person
One who is rachmonah litzlon blind, is exempt from performing keriah, as the obligation is “upon seeing”, and such a person is unable to see (Shu”t Tzitz Eliezer 16:39, and see further Igros Moshe, Orach Chaim Vol. 3, siman 85). Although the Mishnah Berurah (561:7) writes: דמשהגיע לצופים צריך לקרוע אע"פ שעדיין לא ראה אותו – “Upon reaching Tzofim one should perform keriah, even though he can’t yet see”, and according to this a blind person should also be obligated, R’ Sroya Davlitsky writes (Kuntros, Achar Kosleinu, ois 19) the poskim already write that a blind person is exempt from birchas reiyah [blessings related to seeing], and for this reason a blind person is exempt from kiddush levonah (see Birkay Yosef, 224:1), therefore, the same is with performing keriah when seeing the Makom HaMikdosh, since he can’t see he is exempt.
The above makes a big practical difference: Since the reason a blind person is exempt is because he can’t see, even if one isn’t blind, if he makes it that he can’t see the Makom HaMikdosh he has no obligation to perform keriah. Based on this, we mentioned above that the Steipler would close his eyes when entering the Old City, and only open them when he arrived at the Kosel HaMaravi (see Orchos Rabbeinu, Vol. 1, pg. 322, and Vol. 2, pg. 154, see also Shu”t Teshuvos V’Hanhagos 5:165).
